Confidence interval calculator
Enter your sample mean, standard deviation, sample size, and confidence level to get the confidence interval and margin of error around the mean.
95% CI: 48.04 to 51.96
- Lower bound
- 48.04
- Upper bound
- 51.96
- Margin of error
- ± 1.96
- Standard error
- 1.000
Uses the normal (z) approximation: mean ± z × (sd ÷ √n). For small samples (n < 30) a t-interval is slightly wider and more accurate.
How it works
A confidence interval is a range that, under repeated sampling, would contain the true population mean a stated percentage of the time. It is computed as the sample mean plus and minus a margin of error: mean ± z × (standard deviation ÷ √n).
The z multiplier comes from the confidence level: about 1.64 for 90%, 1.96 for 95%, and 2.58 for 99%. Higher confidence widens the interval. The standard error — standard deviation divided by the square root of sample size — shrinks as the sample grows, which is why larger samples give tighter intervals.
The calculator reports the lower and upper bounds, the margin of error, and the standard error. Everything runs in your browser.
Assumptions and limitations
- This uses the normal (z) approximation. For small samples — roughly n under 30 — a t-interval is slightly wider and more accurate, especially when the standard deviation is estimated from the sample.
- It builds an interval for a mean. Proportions, differences, and other statistics use different formulas.
- A 95% confidence interval does not mean a 95% probability the true mean is in this particular interval — it refers to the long-run behavior of the method, a common and important misinterpretation.
Frequently asked questions
What is a confidence interval?
A confidence interval is a range around a sample estimate that quantifies uncertainty about the true population value. A 95% interval is built by a method that, across many samples, captures the true value 95% of the time. Wider intervals mean more uncertainty.
How do I calculate a confidence interval for a mean?
Take the sample mean and add and subtract a margin of error equal to the z-value for your confidence level times the standard error (standard deviation ÷ √n). This calculator does it for you at 90%, 95%, or 99%.
What does a 95% confidence level actually mean?
It refers to the method, not the single interval: if you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true mean. It is not a 95% probability that the true mean lies in this one interval.
When should I use a t-interval instead?
Use a t-interval when the sample is small (roughly under 30) and the standard deviation is estimated from the sample. The t-distribution is slightly wider to account for that extra uncertainty; for large samples the two converge and the z-interval here is fine.
Is my data stored by this calculator?
No. All calculations run client-side in your browser. Nothing you enter is transmitted, logged, or stored.
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